Recursion operator and dispersionless rational Lax representation

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Physics Letters A 297 (2002) 402–407

www.elsevier.com/locate/pla

Recursion operator and dispersionless rational Lax representation

K. Zheltukhin

Department of Mathematics, Faculty of Sciences, Bilkent University, 06533 Ankara, Turkey

Received 19 September 2001; received in revised form 13 February 2002; accepted 27 March 2002 Communicated by A.P. Fordy

Abstract

We consider equations arising from dispersionless rational Lax representations. A general method to construct recursion operators for such equations is given. Several examples are given, including a degenerate bi-Hamiltonian system with a recursion operator.2002 Elsevier Science B.V. All rights reserved.

PACS: 02.30.Ik; 02.30.Sr

Keywords: Integrable system; Recursion operator

1. Introduction

Recently a new method of constructing a recursion operator from Lax representation was introduced in [1]. This construction depends on Lax representation of a given system of PDEs. Let

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Lt= [A, L],

be Lax representation of an integrable nonlinear system of PDEs. Then a hierarchy of symmetries can be given by

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Ltn= [An, L], n = 0, 1, 2, . . .,

where t0= t, A0= A and An, n= 0, 1, 2, . . ., are Gel’fand–Dikkii operators given in terms of L. The recursion

relation between symmetries can be written as

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Ltn+1= LLtn+ [Rn, L], n = 0, 1, 2, . . .,

where Rnis an operator such that ord Rn= ordL.

This symmetry relation allows us to find Rn, hence Ltn+1, in terms of L and Ltn.

In [1,2] this method was applied to construct recursion operators for Lax equations with different classes of scalar and shift operators, corresponding to field and lattice systems, respectively. In [3] the method was applied to

E-mail address: zhelt@fen.bilkent.edu.tr (K. Zheltukhin).

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dispersionless Lax equations on a Poisson algebra of Laurent series (4) Λ= _+∞ −∞ uipi: ui—smooth functions ,

with a polynomial Lax function. The present work is a continuation of [3]. Here we consider a dispersionless Lax equation on the Poisson algebra Λ with a rational Lax function. Such equations one can find in context of topological field theories (see [4,5]).

We have a Lax function

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L=∆1 ∆2 ,

where ∆1, ∆2are polynomials of degree N and M, respectively, and N > M. The dispersionless Lax equation is

(6) ∂L ∂tn= LN−M1 +n 0, L , n= 0, 1, 2, . . .,

where the Poisson bracket is given by

{f, g} = p ∂f ∂p ∂g ∂x− ∂f ∂x ∂g ∂p .

Eq. (6) is of hydrodynamic type. There are several methods for construction of a recursion operator for some equations of hydrodynamic type (see [6–8]). Also a recursion operator can be found with the help of two compatible Hamiltonian formulations of a given equation. For Hamiltonian formulations of equations of hydrodynamic type see Refs. [9,10] and for Hamiltonian formulations of the equations admitting a dispersionless Lax representation see Refs. [11–15].

We construct a recursion operator for a hierarchy of symmetries (6), using a dispersionless Lax representation. First we study the symmetry relation (3) for the rational Lax function. Then we give some examples of calculation of a recursion operator. In particular, we find a recursion operatorR for Eq. (6) with the Lax function

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L= p + S + P p+ Q,

which leads to the system [11]

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St= Px, Pt= P Sx− QPx− P Qx, Qt= QSx− QQx.

The recursion operator is given by

(9) R =   S 1 P Q−1+ PxD_x−1· Q 2P S− Q −2P + (P Sx− (P Q)x)D−1_x · Q Q 1 P Q−1+ S − Q + (QSx− QQx)D−1_x · Q   .

In [11] bi-Hamiltonian representation of this equation was constructed with Hamiltonian operators

(10) D1=  P0 −2P Q −QP Q2 Q −Q2 0   Dx+ ₀ _P x Qx 0 −(P Q)x −QQx 0 −QQx 0 , and D2=   2P P (S− 3Q) Q(S− Q) P (S− 3Q) P2P− 2SQ + 4Q2 Q2P− SQ + Q2 Q(S− Q) Q2P− SQ + Q2 2Q2   Dx

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(11) +   Px SPx− 2(P Q)x SQx− QQx P Sx− (QP )x −SP Q + P2_{+ 2P Q}2 x Qx 2P+ Q2− SQ QSx− QQx Q(2Px+ 2QQx− Sx− SQQx) 2QQx   .

These Hamiltonian operators are degenerate, so, one cannot use them to find a recursion operator. But it turns out that they are related to the recursion operatorR. One can easily check that the following equality holds

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RD1= D2.

We observe that the degeneracy in the bi-Hamiltonian operators is due to the following fact. Let p= p + F then the Lax function becomes

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L= p+ G + P p.

This means that we have two independent variables P and G, where G= S − F . The equation corresponding to the Lax function (13) has been studied in [3].

To remove degeneracy one can take the Lax function as

(14) L= p + S +P p + m i=1 Qi p+ Fi.

As an example we shall consider the Eq. (6) with the Lax function

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L= p + S +P p +

Q p+ F.

2. Symmetry relation for rational dispersionless Lax representation

Following [1] we consider the hierarchy of symmetries for the dispersionless Lax equation (6) with the Lax function (5) (16) ∂L ∂tn= LN−M1 +n 0, L , n= 0, 1, 2, . . .,

Lemma 1. For any n= 0, 1, 2, . . .,

(17) ∂L ∂tn= L ∂L ∂tn−1 + {Rn , L}.

Function Rnhas a form

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Rn= A +

B ∆2

,

where A is a polynomial of degree (N− M) and B is a polynomial of degree (M − 1). Proof. We have LN−M1 +n 0= L LN−M1 +(n−1) 0+ L LN−M1 +(n−1) <0 0. So, LN−M1 +n 0= L LN−M1 +(n−1) 0+ L LN−M1 +(n−1) <0 0− L LN−M1 +(n−1) 0 <0.

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If we take (19) Rn= L LN−M1 +(n−1) <0 0− L LN−M1 +(n−1) 0 <0, then LN−M1 +n 0= L LN−M1 +(n−1) 0+ Rn. Hence, ∂L ∂tn= LN−M1 +n 0, L =L LN−M1 +(n−1) 0+ Rn, L = L∂L ∂tn + {Rn , L},

and (17) is satisfied. The remainder Rnhas form (18). Indeed in (19) we set

A= L LN−M1 +(n−1) <0 0, and B= ∆2· L LN−M1 +(n−1) 0 <0.

Then A is a polynomial of degree (N− M − 1) and B is a polynomial of degree (M − 1). ✷ Now we can apply the Lemma 1 to find recursion operators.

3. Examples

Example 2. Let us consider the Eq. (8) given in introduction. Lemma 3. A recursion operator for (8) is given by (9).

Proof. Using (18) for Rn, we have Rn= A +_p_+QB . So, the symmetry relation (17) is

∂S ∂tn+ ∂P ∂tn · 1 p+ Q+ ∂Q ∂tn · P (p+ Q)2 = p+ S + P p+ Q ∂S ∂tn−1 + ∂P ∂tn−1 · 1 p+ Q+ ∂Q ∂tn−1 · P (p+ Q)2 + p Ax+ Bx p+ Q+ −BQx (p+ Q)2 1+ −P (p+ Q)2 − pB (p+ Q)2 Sx+ Px p+ Q+ −P Qx (p+ Q)2 .

To have the equality the coefficients of p and (p+ Q)−3 must be zero. It gives the recursion relations to find A and B. Then the coefficients of p0, (p+ Q)−1, (p+ Q)−2give expressions for_∂t∂S

n,

∂P ∂tn,

∂Q ∂tn. ✷

Example 4. The dispersionless Lax equation (6) with the Lax function (15), for n= 1, gives the following system

(20)

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Lemma 5. A recursion operator for (20) is given by (21)                 S 2+ PxD_x−1· P−1 1 QF−1+ QxD_x−1· F−1 2P S+ QF−1+ P SxD−1_x · P−1 P F−1 −2P QF−2 + P F−1_Q x− QF−1Fx D−1_x · P−1 − P F−1Qx− QF−1Fx D−1_x · F−1 2Q −QF−1 S− F − P F−1 −2P QF−2− 2Q − P F−1_(Q_x_{− QF}−1_F_x_)D−1 x · P−1 + P F−1 Qx− QF−1Fx D−1_x · F−1 + (QSx− QFx− FQx)D_x−1· F−1 F 1+Px− P F−1Fx D−1_x · P−1 −1 P F−1− F + (FSx− FFx)D_x−1· F−1 −Px− P F−1Fx D−1_x · F−1                 .

Proof. Using (18) for Rn, we have Rn= C +A_p+_p_+FB . So, the symmetry relation (17) is

∂S ∂tn+ ∂P ∂tn · 1 p+ ∂Q ∂tn · 1 (p+ F )+ ∂F ∂tn · −Q (p+ F )2 = p+ S +P p + Q p+ F ∂S ∂tn−1 + ∂P ∂tn−1 · 1 p+ ∂Q ∂tn−1 · 1 (p+ F )+ ∂F ∂tn−1 · −Q (p+ F )2 + p −B p2 + −C (p+ F )2 Sx+ Px p + Qx (p+ F )+ −QFx (p+ F )2 − p Ax+ Bx p + Cx (p+ F ) + −CFx (p+F )2 1+P p + −Q (p+ F )2 .

Therefore, the coefficients of p, p−2and (p+ F )−3must be zero, it gives recursion relations to find A, B and C. Then the coefficients of p0, p−1, (p+ F )−1 and (p+ F )−2, give expressions for_∂t∂S

n, ∂P ∂tn, ∂Q ∂tn and ∂F ∂tn. ✷ Acknowledgements

I thank Professors Metin Gürses, Atalay Karasu and Maxim Pavlov for several discussions. This work is partially supported by the Scientific and Technical Research Council of Turkey.

References

[1] M. Gürses, A. Karasu, V.V. Sokolov, J. Math. Phys. 40 (1999) 6473. [2] M. Blaszak, Rep. Math. Phys. 48 (1-2) (2001) 27.

[3] M. Gürses, K. Zheltukhin, J. Math. Phys. 42 (2001) 1309.

[4] B.A. Dubrovin, Geometry of 2D Topological Field Theories, Lecture Notes in Mathematics, Vol. 1620, Springer, Berlin, 1993, pp. 120– 348.

[5] S. Aoyama, Y. Kodama, Commun. Math. Phys. 182 (1996) 185.

[6] M.B. Sheftel, Generalized hydrodynamic-type systems, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, CRC Press, New York, 1996, pp. 169–189.

[7] V.M. Teshukov, LIIAN 106 (1989) 25.

[8] A.P. Fordy, B. Gürel, Theoret. Math. Phys. (1999).

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[10] E.V. Ferepantov, Hydrodynamic-type systems, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, CRC Press, New York, 1994, pp. 303–331.

[11] I.A.B. Strachan, J. Math. Phys. 40 (1999) 5058.

[12] D.B. Fairlie, I.A.B. Strachan, Inverse Problems 12 (1998) 885.

[13] J.C. Brunelli, M. Gürses, K. Zheltukhin, Rev. Math. Phys. 13 (4) (2001) 529. [14] J.C. Brunelli, A. Das, Phys. Lett. A 235 (1997) 597.